#387970

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,25,22,24,29,31)(19,23,26,30,20,27,28,32);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,22,20)(18,23,24,27)(21,26,29,28)(25,30,31,32);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,20,22,19)(18,23,24,27)(21,26,29,28)(25,32,31,30);(1,5,6,4)(2,7,3,8)(10,14)(11,16)(18,24)(21,29)(23,27)(26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=egi,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295166);